=Paper=
{{Paper
|id=Vol-3183/141
|storemode=property
|title=Instantaneous Real-Time Kinematic Decimeter-level Positioning with Galileo and BDS-3 Penta-Frequency Signals Over Long-Baseline
|pdfUrl=https://ceur-ws.org/Vol-3183/paper10.pdf
|volume=Vol-3183
|authors=Liwei Liu,Shuguo Pan,Wang Gao,Chun Ma
|dblpUrl=https://dblp.org/rec/conf/icl-gnss/LiuPGM22
}}
==Instantaneous Real-Time Kinematic Decimeter-level Positioning with Galileo and BDS-3 Penta-Frequency Signals Over Long-Baseline==
https://ceur-ws.org/Vol-3183/paper10.pdf
Instantaneous Real-Time Kinematic Decimeter-Level Positioning
with Galileo and BDS-3 Penta-Frequency Signals Over Long-
Baseline
Liwei Liua,b, Shuguo Pana,b , Wang Gaoa,b and Chun Ma a
a
School of Instrument Science and Engineering, Southeast University, Sipailou 2, Nanjing, 210096, China
b
Key Laboratory of Micro-Inertial Instrument and Advanced Navigation Technology, Ministry of 7 Education,
Sipailou 2, Nanjing, 210096, China
Abstract
To take full advantages of Galileo and BDS-3 penta-frequency signals, a long-baseline RTK
positioning method based on Galileo and BDS-3 penta-frequency ionosphere-reduced (IR)
combinations is proposed. First, the high-quality signals with low-noise and weak-ionospheric
delay characteristics of Galileo and BDS-3 are analyzed. Second, the multi-frequency extra-
wide-lane (EWL)/ wide-lane (WL) combinations with long-wavelengths are constructed. Third,
the IR-EWL combinations are calculated by geometry-free (GF) method, then the resolved IR-
EWL combinations are used to constrain the IR-WL, of which the ambiguities can be obtained
in a single epoch. There is no need to consider the influence of ionospheric parameters in the
third step because the ionospheric delay factors of IR-EWL/WL combinations are close to 0.
Compared with the estimated ionosphere model, the proposed method can improve the
availability of positioning and reduce the number of parameters by half and the required
operation time is greatly reduced. Therefore, it reduces the dimension of parameter estimation
and is suitable for the use of multi-frequency and multi-system real-time RTK. The results
using real data show that stepwise fixed model of the IR-EWL/WL combinations can realize
long-baseline instantaneous decimeter-level positioning.
Keywords 1
Galileo, BDS-3, Penta-frequency, Ionosphere-reduced, RTK positioning, Long-baseline
1. Introduction
European Galileo Satellite Navigation System and China Beidou-3 Global Satellite Navigation
System have been launched and broadcast penta-frequency signals. At present, Galileo broadcasts E1
(1575.42 MHz), E5a (1176.45 MHz), E5b (1207.14 MHz), E5 (1191.795 MHz) and E6 (1278.75 MHz)
and BDS-3 broadcasts B1C (1575.42 MHz), B1I (1561.098 MHz), B3I (1268.52 MHz), B2a (1176.45
MHz), B2b (1207.14 MHz) [1]. Taking advantage of multi-system and multi-frequency signals, in
addition to increasing observation redundancy, it can also construct some observation value
combinations with excellent characteristics such as long wavelength, weak ionospheric delay and low
noise factor. And the performance of the positioning solution can be greatly improved [2].
The application of multi-frequency signals combinations was proposed by Forssell and Jung. The
basic principle is the classical GF method to fix the ambiguities of EWL, WL and narrow lane (NL)
step by step according to the difficulty of ambiguity resolution [3]. Feng researched and gave the integer
combination coefficients to reduce the influence of the ionosphere in the process of ambiguity
resolution at all levels of GPS/Galileo/BDS-2 EWL/WL/NL combinations [5]. Gao studied the
ionosphere-reduced NLs of BDS-3 and Galileo penta-frequency, the model strength and positioning
ICL-GNSS 2022 WiP, June 07–09, 2022, Tampere, Finland
EMAIL: liuliwei@seu.edu.cn (L. Liu); psg@seu.edu.cn (S. Pan); gaow@seu.edu.cn (W. Gao) ; machun@seu.edu.cn (C. Ma)
ORCID: 0000-0002-1151-8820 (L. Liu); 0000-0003-0724-9020 (S. Pan); 0000-0002-7782-9533 (W. Gao); 0000-0003-0841-7540 (C. Ma);
©️ 2022 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org) Proceedings
accuracy of ambiguity resolution are improved compared with the traditional dual-frequency
ionosphere-free combination [6]. Based on the new frequency signals of Galileo and BDS-3, we studied
the selection criteria for the combination of ionosphere-reduced observations suitable for long baselines.
The optimal ionosphere-reduced combinations of Galileo and BDS-3 penta-frequency signals are
analyzed. The estimated ionosphere model is used to compare the positioning performance with the IR
model, and the decimeter-level positioning of Galileo and BDS-3 penta-frequency minimum number
of parameters to be estimated for long-baseline RTK positioning is realized.
The purpose of this article is to study multi-frequency combination signals of Galileo and BDS-3
with ionospheric delay factor close to 0 and low combined observation noise. With the comparation of
IR model and estimated ionosphere model, positioning performance and positioning efficiency of IR
model is studied. In the following section, we define the conditions to be met by IR-EWL/WL
combinations, and provide the geometric model required by the algorithm. Then experiments were
conducted using a set of long baseline data. Finally, the experimental results are analyzed and
summarized.
2. Penta-frequency Observation Combination Model of Galileo and BDS-3
2.1. Double Difference (DD) Mathematical Model
Ignore satellite systems, the observation equation of the basic pseudo-range and carrier-phase
observations is:
Pi = + i I1 + T + P (1)
i = − i I1 + T + i Ni +
where, the symbol " " represents DD operation; Pi and i represent pseudo-range and carrier
observations, respectively; is the geometric distance between the satellite and the receiver; I1 is the
first-order ionospheric delay at the first frequency; i is the first order ionospheric scale factor; T
indicates tropospheric delay; P and are observation noise of pseudo-range and carrier-phase
respectively; N indicates integer ambiguity; is the carrier wavelength.
Correspondingly, the DD observation equation of the penta-frequency signals after basic observation
equation linear combination is:
(i , j , k , m, n) = − (i , j , k , m, n) I1 + T + (i , j , k , m, n) N(i , j ,k ,m,n ) + (i , j ,k ,m,n ) (2)
where, each parameter is expressed as:
i f1 1 + j f 2 2 + k f3 3 + m f 4 4 + n f5 (3)
(i , j , k , m, n) =
i f1 + j f 2 + k f3 + m f 4 + n f5
f2
( i f1 + j f 2 + k f3 + m f 4 + n f5 ) (4)
( i , j , k , m , n ) = 1
i f1 + j f 2 + k f 3 + m f 4 + n f 5
c (5)
(i , j , k , m, n) =
i f1 + j f 2 + k f3 + m f 4 + n f5
(if1 ) 2 + ( jf 2 ) 2 + (kf3 ) 2 + (mf 4 ) 2 + ( nf5 ) 2 (6)
( i , j , k , m , n ) =
f(i , j , k , m, n)
where, i, j, k , m, n is the combination coefficient; Correspondingly, the calculation of pseudo-range
combination P(i , j , k , m, n ) is similar to that of (i , j , k , m, n ) ; (i , j , k , m, n) is the ionospheric scalar factor of the
combined signal; (i , j , k , m, n) is the wavelength of combined observations; (i , j , k , m, n ) is the noise coefficient
of combined observations; c is the speed of light.
2.2. Selection of optimal IR-EWL combination
The combination of penta-frequency EWL combination can construct infinite combined
observations according to different coefficient values, but most signals do not have the characteristics
of low-noise and weak-ionospheric scale factor. Referring to literature [6], this paper selects the IR-
EWL/WL combinations for long-baseline positioning, and makes the following constraints on the
characteristics of combined observations:
(1) The influence of ionospheric delay on ambiguity resolution is less than 0.02 cycles in unit, which
can be expressed as:
f 2 ( i f1 + j f 2 + k f3 + m f 4 + n f5 ) (7)
( i , j , k , m, n ) = 1
c
The ionospheric delay corresponding to 5 m has less than 0.1 cycles on ambiguity resolution. If the
impact on ranging is less than 5 cm, it is required. In fact, the ionospheric residuals for the 100-500 km
baseline double-differences are less than 1.5m [5].
(2) If the combined noise is required to be small, the combined coefficient of the combination value
should not be too large. Taking the GPS EWL combination (1, 6, -5) as the reference (103.80), the noise
amplification coefficient shall not be greater than 110;
(3) The wavelength of the combined observation value shall not be too small or too large. The
combination wavelength shall be between 0.8m and 10m with reference of GPS triple-frequency WL
combination the twice (1, -1, 0) and (0, 1, -1).
Based on the above three conditions, take [-10, 10] as the search interval of combination coefficient,
the characteristics of combinations meeting the above conditions are shown in Table 1. The sequence
of corresponding Galileo and BDS-3 signal types in the Table 1 is: E1/E5a/E5b/E5/E6 and
B1C/B1I/B3I/B2a/B2b. It can be seen from the table that there are five EWL/WL combinations of BDS-
3 and four EWL/WL combinations of Galileo that meet the conditions, of which BDS-3 (-1, 2, -4, 1, 2)
and Galileo (1, 4, 1, -3, 3) are EWL combinations and the rest are WL combinations. Therefore (-1, 2,
-4, 1, 2) of BDS-3 and (1, 4, 1, -3, 3) of Galileo are selected as the optimal IR-EWL combinations.
Table 1
Ionosphere-reduced WL combinations for BDS-3 and Galileo
Combination (i , j , k , m, n) (i , j , k , m, n ) cycle m−1 (i , j , k , m, n) ( i , j , k , m , n )
Coefficient /m /
BDS-3
(-1, 2, -4, 1, 2) 4.7266 -0.0023 -0.0005 105.9863
(-1, 3, -6, 6, -2) 2.1236 0.0058 0.0027 83.2113
(6, -4, -7, 6, -1) 1.6651 -0.0048 -0.0029 79.9460
(-5, 8, -9, 8, -2) 1.5588 -0.0094 -0.0060 109.4091
(-2, 5, -10, 7, 0) 1.4653 0.0033 0.0023 84.5970
Galileo
(1, 4, 1, -3, -3) 3.9074 -0.0012 -0.0003 66.5653
(2, 7, 1, -4, -6) 1.9537 -0.0057 0.0029 89.6363
(3, 2, -2, 7, -10) 1.3630 0.0094 0.0069 74.2670
(3, 1, -3, 9, -10) 1.3630 0.0062 0.0046 80.7461
2.2.1. Selection of optimal IR-EWL combinations using GF model
Equation (8) calculates the IR-WL by using the combination of GIF:
P[ a ,b,c , d ,e ] − (i , j , k , m, n) (8)
N(i , j , k , m, n) =
(i , j , k , m, n)
P[a,b,c,d ,e] = aP1 + bP2 + cP3 + d P4 + eP5 (9)
where, • represents the rounding operator, P[ a,b,c, d ,e] is the linear combination form of DD
observations of pseudo-range combination, and the combination coefficient of pseudo-range a, b, c, d , e
is any real number. As shown in equation (10-12), considering that the sum of pseudo-range coefficients
is 1, the sum of ionospheric scale factor of combined pseudo-range observations and IR is 0, and the
combined noise is the smallest, the optimal pseudo-range coefficient can be calculated by the minimum
norm method, as shown in Table 2. It should be noted that only up to five significant figures are
displayed in the table.
a +b+c+ d +e =1 (10)
f12
f12
f1 2
f12
(11)
( i , j , k , m , n ) + a + b +c +d + e =0
f 22 f32 f 42 f52
a2 + b2 + c2 + d 2 + e2 = min (12)
(if1 ) + ( jf 2 ) + (kf 3 ) + ( mf 4 ) + ( nf 5 ) 2
2 2 2 2 2 (13)
+ (a 2 + b 2 + c 2 + d 2 + e 2 )
2
P
f (2i , j , k , m , n )
N =
( i , j ,k ,m ,n ) (i , j , k , m , n )
where, and P represent the DD noise of non-combined carrier observation value and pseudo-
range observation respectively, and the values in this paper are 0.5 m and 5 mm respectively.
The optimal pseudo-range coefficient combination of each IR combination is brought into
respectively, and the rounding success rate of ambiguities of IR-WL Ps is calculated by equation (14)
[8].
1 ( x − )2 (14)
Ps ( −0.5 x ) =
0.5
exp − dx
−0.5
2 2 2
Kijkmn, abdce I (15)
GF =
(i , j , k , m, n )
, Kijkmn, abdce = (i, j , k , m, n) + ( a,b,c, d ,e)
where, GF is the systematic deviation caused by unmodeled errors. When calculating the ambiguities
of GIF EWL combinations, the first-order ionospheric delay, tropospheric delay and satellite orbit error
are eliminated. Therefore, the ambiguity accuracy is only affected by observation noise and second-
order ionospheric delay. Here, the influence of second-order ionospheric delay can be ignored, so GF
can be regarded as 0, while Kijkmn,abdce = 0 .
It can be seen from Table 2 that BDS-3 (-1, 2, -4, 1, 2) combination can obtain a rounding success
rate of 100% and Galileo (1, 4, 1, -3, 3) combination can obtain a rounding success rate of 99.56%.
Therefore, (-1, 2, -4, 1, 2) and (1, 4, 1, -3, 3) combination is only affected by pseudo-range noise, and
under the condition of good observation accuracy, IR-EWL combinations can be fixed by rounding in
a single epoch.
Table 2
Optimal combination of pseudo-range coefficients of ionosphere-reduced WL combinations and
success rate by rounding
Ambiguity Success Rate
Combination Pseudo-range Coefficient Combination
Accuracy by
Coefficient a, b, c, d , e
/cycle Rounding/%
BDS-3
(-1, 2, -4, 1, 2) 1.2140 1.1685 -0.1227 -0.7409 -0.5190 0.103 100.0
(-1, 3, -6, 6, -2) 1.2198 1.1742 -0.1245 -0.7463 -0.5232 0.228 97.16
(6, -4, -7, 6, -1) 1.2122 1.1668 -0.1221 -0.7392 -0.5178 0.290 91.50
(-5, 8, -9, 8, -2) 1.2089 1.1637 -0.1210 -0.7361 -0.5154 0.310 89.35
(-2, 5, -10, 7, 0) 1.2180 1.1724 -0.1239 -0.7446 -0.5219 0.330 86.98
Galileo
(1, 4, 1, -3, -3) 2.2153 -0.6790 -0.3506 -0.5116 0.3260 0.175 99.56
(2, 7, 1, -4, -6) 2.2094 -0.6765 -0.3490 -0.5096 0.3256 0.351 84.63
(3, 2, -2, 7, -10) 2.2153 -0.6790 -0.3506 -0.5116 0.3260 0.351 84.59
(3, 1, -3, 9, -10) 2.2250 -0.6833 -0.3532 -0.5150 0.3266 0.504 67.92
2.2.2. Selection of optimal IR-EWL combinations using GB model
Selecting IR-EWL combinations can make full use of pseudo-range observation data while using
GB model. Referring to literature [7], using the estimated ionosphere model to calculate EWL
combinations. The corresponding model is:
1 1 0 P2 (16)
1
2 0
P
2
1 3 0 P2
A= , R=
1 4 0 P2
1 5 0 P2
1 − EWL EWL EWL
2
( )
−1
P = R −1 , Q = AT PA (17)
where, A is the design matrix, and the corresponding estimation parameters are station satellite
distance accuracy, ionosphere accuracy and ambiguity accuracy; R is the corresponding observation
noise variance covariance matrix; EWL2
= EWL
2
2 is the observation noise of EWL; The matrix Q is the
variance covariance matrix of parameter estimation, which reflects the accuracy of parameter estimation,
and its diagonal element is the variance of estimated parameters. The IR-EWL combinations is affected
by the unmodeled atmospheric residual and orbit error when GB model used. Specifically, in
equation (14) can be calculated by equation (18):
orb + T − (i , j , k , m.n ) I (18)
GB =
(i , j , k , m.n )
Referring to literature [5], it is assumed that under the conditions of medium and long-baseline, the
tropospheric residuals are 10 cm and 15 cm respectively, and the first-order ionosphere residuals are 80
cm and 100 cm respectively. The corresponding ambiguity accuracy and success rate used GB model
are shown in Table 3.
Table 3
BDS-3 and Galileo optimal GB-IR ambiguity accuracy and success rate by rounding
Combination Ambiguity Success Rate by Rounding/% cm
Coefficient Accuracy /cycle T =0, I =0 T =10, I =80 T =15, I =100
BDS-3
(-1, 2, -4, 1, 2) 0.232 96.91 96.81 96.71
(-1, 3, -6, 6, -2) 0.494 68.84 68.49 68.16
(6, -4, -7, 6, -1) 0.634 56.97 56.69 56.41
(-2, 5, -10, 7, 0) 0.705 52.16 51.93 51.70
(-5, 8, -9, 8, -2) 0.717 51.44 51.14 50.85
Galileo
(1, 4, 1, -3, -3) 0.333 86.68 86.52 86.36
(2, 7, 1, -4, -6) 0.653 55.62 55.43 55.24
(3, 2, -2, 7, -10) 0.934 40.74 40.56 40.38
(3, 1, -3, 9, -10) 0.940 40.52 40.35 40.18
Regarding of estimated ionosphere method, the GB model reduces the model strength, which is
equivalent to the ionosphere-fixed model [8]. Although the BDS-3 IR-EWL combination (-1, 2, -4, 1,
2) or Galileo IR-EWL combination (1, 4, 1, -3, -3) cannot obtain 100% success rate by using GB model,
if four linearly independent EWL combinations are found, the ambiguities of IR-EWL combinations
can be obtained by using linear combinations. Unfortunately, only three groups of linearly independent
EWL combinations with high ambiguity accuracy can be obtained. Therefore, the linear combination
method is not suitable to provide the success rate of IR-EWL combinations.
2.3. Selection of calculation model for EWL/WL combinations
Therefore, the IR-EWL combinations can be obtained directly by rounding using GF model.
In this paper, the IR-EWL combinations of BDS-3 or Galileo is calculated by GF method. As a
comparison, with reference to equation (19), which estimates ionospheric parameters with low-noise
EWL combinations:
vP B I 0 lP
(19)
1 1
1 s
2 I s
P2 B
v 0 x P2
l
v l
P3 = B 3 I s 0
ion −
P3
vP B 4 I s 0 N l P4
4 EWL l
v P5 B 5 I s 0 P5
B − EWL I s EWL I s
v EWL l EWL
After fixing EWL, the calculation of WL ambiguity is selected according to its combination
characteristics. Generally, the fixed EWL combinations are used to restrict the ambiguity of WL
combinations, as shown in equation (20).
x (20)
v EWL
'
B − EWL I s 0 l EWL
'
= ion −
vWL B −WL I s WL I s lWL
NWL
The WL is constrained with IR-EWL by GB model, and ionosphere can be ignored because it is
very little. Its estimation equation is:
v EWL
'
B 0 x l EWL
'
(21)
= −
vWL B WL I s NWL lWL
It can be seen that using the IR-EWL combination to restrict the IR-WL combinations does not need
to estimate the ionospheric parameters, and the dimension of parameter estimation can be reduced.
2.4. Selection of IR-WL combinations
Based on equation (21), calculate the DD float ambiguities of the remaining four IR-WL combinations
in Table 1 after ambiguities of BDS-3 IR-EWL combination (-1, 2, -4, 1, 2) and Galileo IR-EWL
combination (1, 4, 1, -3, -3) are fixed. The solution accuracy and the fixed DD ranging accuracy are
shown in Table 4.
Table 4
Ambiguity accuracy of IR-WL combinations constrained by fixed EWL combination BDS-3 IR-EWL
combination (-1, 2, -4, 1, 2) and Galileo IR-EWL combination (1, 4, 1, -3, -3)
Combination Coefficient (i , j , k , m, n) /m WL/cycle DD Range/m
BDS-3
(-1, 3, -6, 6, -2) 2.124 0.161 0.381
(6, -4, -7, 6, -1) 1.665 0.360 0.342
(-5, 8, -9, 8, -2) 1.559 0.212 0.451
(-2, 5, -10, 7, 0) 1.465 0.161 0.381
Galileo
(2, 7, 1, -4, -6) 1.954 0.047 0.317
(3, 1, -3, 9, -10) 1.363 0.408 0.318
(3, 2, -2, 7, -10) 1.363 0.362 0.319
It can be seen that the accuracy of WL combination Galileo (2, 7, 1, -4, -6) is the best and the float
accuracy of WL is also the best, so it is selected as the optimal IR-WL combination of Galileo. DD
Range of BDS-3 (6, -4, -7, 6, -1) is best after fixed, but the float accuracy of which is the worst, while
the float accuracy of WL combination BDS-3 (-1, 3,-6, 6, -2), BDS-3 (-2, 5, -10, 7, 0) are best. Although
DD range of WL combination BDS-3 (-1, 3,-6, 6, -2), (-2, 5, -10, 7, 0) is not optimal, it is close to
optimal. In addition, the ionospheric scale factor of WL combination BDS-3 (-1, 3, -6, 6, -2) is smaller,
so it is selected as the optimal IR-WL combination of BDS-3.
3. Experiment and analysis
In this paper, a group of 189.4 km long-baseline of IGS station are used for the experiment. The data
comes from TIT2 and FFMJ stations of BKG data center. The observation date is UTC time, October
1, 2021 (24 hours), day of year is 274, and the sampling interval is 30 s. During the calculation, the cut-
off angle of the satellite in the calculation is set to 15°.
The number of BDS-3 and Galileo satellites with five frequencies and their RDOP in this period are
shown in Figure 1. The number of common view satellites of the two stations fluctuates in the range of
8-14 mostly. In the 2279th epoch, rdop increased sharply due to the small number of visible satellites.
Figure 2 shows the sky plots of BDS-3 and Galileo various satellites in the experiment.
Figure 1: Number of satellites and RDOP value of BDS-3 and Galileo in full time
(a) (b)
Figure 2: Sky plots for the various satellites of BDS-3 (a) and Galileo (b)
Figure 3 and Figure 4 shows the fraction bias of EWL combinations ambiguities using estimated
ionosphere model and IR-GF model, respectively. Different colors correspond to different satellite pairs.
It can be seen that the estimated ionosphere model can be all within 0.25 cycles and can be reliably
rounded and fixed.
However, either BDS-3 or Galileo, the accuracy of IR-EWL ambiguities calculating by GIF method
is poor because of greater pseudo-range noise, which affects the ambiguity accuracy. Therefore, the
influence of the ionosphere can be properly ignored to reduce the noise of pseudo-range observations.
This paper only analyzes the case without pseudorange noise reduction.
(a) (c)
(b) (d)
Figure 3: The fractions of EWL ambiguity of IR-GF Figure 4: The fractions of EWL ambiguity of
model: BDS-3 (a) and Galileo(b) estimated ionosphere model: BDS-3 (c) and
Galileo(d)
The true values of IR-EWL ambiguities are all obtained by multi-epoch filtering. It can be seen from
the Table 5 that although the accuracy of IR-EWL ambiguity calculated by GIF model is not high, the
rounding reaches 99.04% (BDS-3) and 97.89% (Galileo). In the experiment, the threshold of decimal
deviation is set as 0.3 when rounding EWL.
Table 5
The ambiguity accuracy of BDS-3 and Galileo IR-EWL combinations calculated by pseudo-range with
optimal coefficients.
Noise
Pseudo-range Coefficients EWL Fractions/cycle %
Factor
a b c d e 0.5 0.5 0.4 0.3 0.2
BDS-3
1.214 1.169 -0.123 -0.741 -0.519 1.916 0.96 99.04 97.06 91.02 75.58
Galileo
2.215 -0.679 -0.351 -0.512 0.326 2.420 2.11 97.89 94.64 86.77 70.07
The WL ambiguities are calculated by IR-GB model and estimated ionosphere model and the
suboptimal/optimal ambiguity variance ratio (Ratio value) is shown in Figure 5 and Figure 6. In the
figure, the values corresponding to the red lines in the upper and lower figures are 10 and 2.5
respectively. It can be seen that ratio value of IR-GB model is greater than that of estimated ionosphere
model. The reason is that IR-GB model does not need to estimate the ionospheric delay, so the strength
of the parameter estimation model is greater.
Figure 5: The Ratio of WL ambiguity in IR-GB Figure 6: The Ratio of WL ambiguity in
model estimated ionosphere model
The threshold for LAMBDA estimation of WL ambiguities is 0.2 cycles. If the number of WL
ambiguity float solutions satisfying the condition is less than 4, the positioning result of this epoch is
considered invalid. After the WL ambiguities are fixed, the observation equations are brought back to
obtain the coordinate solution under the fixed solution. The positioning error of the corresponding
solution coordinates in the East (E), North (N) and Up (U) directions is shown in Figure 7 and Figure
8.
Figure 7: The positioning results in IR-GB model Figure 8: The positioning results in estimated
ionosphere model
Finally, the positioning accuracy statistics of the two methods are shown in Table 6. It can be seen
that the accuracy of the IR-GB model and that of the estimated ionosphere model is basically the same.
However, the IR does not need to estimate the ionospheric delay term, so it can achieve higher
ambiguity calculation efficiency. Especially in the multi-level step-by-step resolution of multi-system
ambiguity, it will be more obvious. Figure 9 and Table 7 shows the operation time of these two models.
It can be seen that the operation time of the IR model is significantly lower than that of estimated
ionosphere model, which is consistent with the analysis.
Table 6
Statistics of the positioning results with EWL/WL observations
Positioning Model N/m E/m U/m Positioning success rate %
Estimated ionosphere 0.159 0.206 0.408 98.4
Ionosphere-reduced 0.168 0.191 0.375 99.5
Figure 9: Operation time of the IR model (red) and estimated ionosphere model (blue)
Table 7
Statistics of operation time of the IR model and estimated ionosphere model
Positioning Model Mean/ms Max/ms Min/ms
Estimated Ionosphere 13.8 67 1
Ionosphere-reduced 3.7 18 <1
4. Conclusions
In this paper, a step-by-step method for fixing the ambiguities of IR-EWL/WL combinations is
proposed. First, the IR-EWL combinations are calculated by GF method, then the fixed IR-EWL
combinations are used to constrain the IR-WL combinations, of which the ambiguities can be obtained
in a single epoch. The proposed IR model does not need to estimate the ionospheric delay, so the
strength of the parameter estimation model is greater.
The main difference between the ionosphere-reduced model and estimated ionospheric model is the
accuracy of the WL float ambiguities. Once the EWL of the IR model is successfully fixed, a positioning
performance comparable to that of the estimated ionosphere model can be obtained.
The experiment shows that the positioning performance of the IR model is comparable to that of
estimated ionosphere model (0.168/0.191/0.375m vs. 0.159/0.206/0.408m), and positioning
performence is more effective (99.5% vs. 98.4%) and the computation time is shorter (3.7ms vs.
13.7ms).
The performance of the ratio depends on the strength of the model. The ionosphere-reduced model
is essentially an ionospheric-fixed model, which reduces the number of parameters to be estimated to
improve the model and obtain a higher-precision float solutions.
5. Acknowledgements
IGS MGEX is gratefully acknowledged for providing Galileo and BDS-3 data. This research is
supported by the National Key R&D Program of China (No. 2021YFC3000502); the National Natural
Science Foundation of China (No. 41904022); and the Foundation of Laboratory of Science and
Technology on Marine Navigation and Control, China State Shipbuilding Corporation (No.
2021010104).
6. References
[1] Yang, Y., Mao, Y., Sun B.: Basic performance and future developments of BeiDou global
navigation satellite system. Satellite Navigation, 1(1), 1, (2020). doi:10.1186/s43020-019-0006-0
[2] Zhang, Z., Li, B., He, X.: Multi-frequency phase ambiguity of Beidou-3 fixed method without
geometric single epoch. Journal of Surveying and Mapping, 49(9), 1139-1148, (2020). URL:
http://dx.doi.org/10.11947/j.AGCS.2020.20200325
[3] Forssell, B., Martin-Neira, M., Harrisz, R.: Carrier phase ambiguity resolution in GNSS-2.
Proceedings of the 10th International Technical Meeting of the Satellite Division of the Institute
of Navigation, 1727-1736 (1997).
[4] Jung, J., Enge, P., Pervan, B.: Optimization of cascade integer resolution with three civil GPS
frequencies. Proceedings of the 13th International Technical Meeting of the Satellite Division of
The Institute of Navigation, 2191-2200 (2000).
[5] Feng, Y.: GNSS three carrier ambiguity resolution using ionosphere-reduced virtual signals.
Journal of Geodesy, 82(12), 847-862, (2008). URL: https://doi.org/10.1007/s00190-008-0209-x
[6] Gao, W., Pan, S., Huang G.: Long-term baseline RTK positioning method based on the
combination of BDS-3 and Galileo multi-frequency signal weak ionosphere. Journal of Chinese
Inertial Technology, 28(6), 6, (2020). doi: 10.13695/j.cnki.12-1222/o3.2020.06.013
[7] Laurichesse, D.; Banville, S. Innovation: Instantaneous centimeter-level multi-frequency precise
point positioning. GPS World J. 29, 43–47, (2018).
[8] Li, B., Feng, Y., Gao, W.: Real-time kinematic positioning over long baselines using triple-
frequency BeiDou signals. IEEE Transactions on aerospace and electronic systems, 51(4), 3254-
3269, (2015). doi: 10.1109/TAES.2015.140643